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Systems & Control Letters 51 (2004) 57–65www.elsevier.com/locate/sysconle

Delay-dependent robust stability criteria for uncertain neutralsystems with mixed delays

Yong Hea ;∗, Min Wua, Jin-Hua Sheb, Guo-Ping Liuc;d

aSchool of Information Science and Engineering, Central South University, Changsha 410083, ChinabDepartment of Mechatronics, School of Engineering, Tokyo University of Technology, Tokyo 192-0982, Japan

cSchool of Mechanical, Materials, Manufacturing Engineering and Management, University of Nottingham,Nottingham NG7 2RD, UK

dInstitute of Automation, Chinese Academy of Sciences, Beijing 100080, China

Received 18 December 2002; received in revised form 17 June 2003; accepted 30 June 2003

Abstract

This paper concerns the problem of the delay-dependent robust stability of neutral systems with mixed delays andtime-varying structured uncertainties. A new method based on linear matrix inequalities is presented that makes it easy tocalculate both the upper stability bounds on the delays and the free weighting matrices. Since the criteria take the sizes ofthe neutral- and discrete-delays into account, it is less conservative than previous methods. Numerical examples illustrateboth the improvement this approach provides over previous methods and the reciprocal in;uences between the neutral-and discrete-delays.c© 2003 Elsevier B.V. All rights reserved.

Keywords: Neutral system; Delay-dependent criteria; Robust stability; Time-varying structured uncertainties; Linear matrix inequality

1. Introduction

There are two types of time delay systems: retarded and neutral. The retarded type [4] contains delays onlyin its states (this kind of delay is called a discrete delay hereafter), whereas the neutral type contains delaysboth in its states and in the derivatives of its states. Neutral-type delay systems can be found in such placesas population ecology, distributed networks containing lossless transmission lines, heat exchangers, robots incontact with rigid environments, etc (e.g., [14–16,20]). Some new control technologies, like repetitive control,use the neutral type through the insertion of an artiBcial neutral delay into a control loop in order to boost thecontrol performance for periodic signals [10]. While the number of unstable poles is Bnite in a retarded-delaysystem, it is inBnite in a neutral-delay system. That makes a neutral-delay system much harder to stabilize,and a considerable number of studies have been carried out on the stability problem of the neutral type.

Although frequency-domain methods, such as the use of the Nyquist stability criterion, are very eCectivein the analysis and synthesis of a control system, they cannot easily handle a delay system that contains

∗ Corresponding author. Tel.: +86-731-887-7354; fax: +86-731-887-9361.E-mail address: heyong08@yahoo.com.cn (Y. He).

0167-6911/$ - see front matter c© 2003 Elsevier B.V. All rights reserved.doi:10.1016/S0167-6911(03)00207-X

58 Y. He et al. / Systems & Control Letters 51 (2004) 57–65

time-varying structured uncertainties (see [2,7]). Instead, the Lyapunov-functional approach is widely employedbecause it is known to be eCective; and linear matrix inequalities (LMIs [1]) provide a powerful and eJcientnumerical tool for the calculations. The stability criteria thus obtained can be divided into two categories:delay independent [2,11,12,19] and delay dependent [3,5,8,9,13,18,17,21–23]. The delay-independent type isindependent of delay size; and as might be expected, it is generally conservative, especially when a delay isshort. On the other hand, studies of delay-dependent criteria have focused mainly on identical delays in neutraland discrete terms [3,5,9,18,17,22]. Some papers have also presented criteria that depend only on the size ofdiscrete delays, and not on the size of neutral delays [8,13,21,23]. They are called discrete-delay-dependentand neutral-delay-independent stability criteria [8]. Although a great deal of eCort has been devoted to theinvestigation of this subject, our knowledge of neutral- and discrete-delay-dependent stability criteria is stillinsuJcient.

Recently, Park [24] presented a new method of obtaining a delay-dependent criterion for a discrete-delaysystem, which was less conservative than previous methods. It was later extended to a neutral-delay system[22,23]. However, that analysis used the Leibniz–Newton formula in the derivative of a Lyapunov–Krasovskiifunctional, and replaced the term x(t − �) with x(t) − ∫ t

t−� x(s) ds in some places, but not in others, in orderto make the Lyapunov–Krasovskii functional easier to handle. Clearly, there must be a relationship betweenx(t − �) and x(t) − ∫ t

t−� x(s) ds because both of them aCect the result; but it was not taken into account. Inaddition, even though the free weighting matrices for those terms are very important, no method of selectingthem was given.

This paper presents a new method of dealing with the problem of the delay-dependent stability of neutralsystems. First, a criterion for a nominal neutral system is derived. This method employs free weightingmatrices to express the in;uences of, and the relationship between, the terms x(t − �) and x(t)− ∫ t

t−� x(s) ds.The new criterion is based on LMIs, which makes the free weighting matrices easy to select. Since thiscriterion is both neutral-delay-dependent and discrete-delay-dependent, it is less conservative than previousmethods for mixed neutral- and discrete-delays. The criterion obtained is then extended to a neutral systemwith time-varying uncertainties. Finally, some numerical examples illustrate both the improvement the proposedapproach provides over previous methods and also the reciprocal in;uences between neutral and discrete delays.

2. Notation and preliminaries

Consider the time-varying structured uncertain neutral system

�:

{x(t) − Cx(t − �2) = (A + QA(t))x(t) + (B + QB(t))x(t − �1); t ¿ 0;

x(t) = �(t); t ∈ [ − �; 0];(1)

where x(t)∈Rn is the state vector; �1; �2 ¿ 0 are constant delays; � := max(�1; �2), A; B; C ∈Rn×n areconstant matrices; and the spectrum radius of the matrix C, �(C), satisBes �(C)¡ 1. The initial condition �(t)denotes a continuous vector-valued initial function of t ∈ [ − �; 0]. The time-varying structured uncertaintiesare of the form

[QA(t) QB(t)] = DF(t)[Ea Eb]; (2)

where D; Ea; Eb are constant matrices with appropriate dimensions; and F(t) is an unknown, real, and possiblytime-varying matrix with Lebesgue measurable elements, and its Euclidean norm satisBes

‖F(t)‖6 1; ∀t: (3)

First, the nominal system �0 of � is deBned to be

�0:

{x(t) − Cx(t − �2) = Ax(t) + Bx(t − �1); t ¿ 0;

x(t) = �(t); t ∈ [ − �; 0]:(4)

Y. He et al. / Systems & Control Letters 51 (2004) 57–65 59

To obtain the main results, the following lemma is necessary to deal with the uncertainties.

Lemma 1 (Xie [25]). Given matrices Q = QT; H; E and R = RT ¿ 0 of appropriate dimensions, then

Q + HFE + ETFTHT ¡ 0

for all F satisfying FTF6R, if and only if there exists an �¿ 0 such that

Q + �2HHT + �−2ETRET ¡ 0:

3. Main results

In order to simplify the treatment of the problem, the operator D :C([− �2; 0];Rn) → Rn is Brst deBned tobe

Dxt = x(t) − Cx(t − �2):

The stability of D is deBned as follows:

De�nition 1 (Hale [6]): The operator D is said to be stable if the zero solution of the homogeneous diCerenceequation

Dxt = 0; t¿ 0; x0 = ∈{�∈C([ − �2; 0]: D� = 0}is uniformly asymptotically stable.

The necessary condition for the stability of � and �0 is that the operator D be stable [6]. The followingtheorem governs the nominal system �0:

Theorem 1. Given scalars �1 ¿ 0 and �2 ¿ 0, the nominal system �0 is asymptotically stable if the operatorD is stable and there exist positive de>nite matrices P = PT ¿ 0, Qi = QT

i ¿ 0 (i = 1; 2) and R = RT ¿ 0,non-negative de>nite matrices Xii = X T

ii ¿ 0 and Yii = Y Tii ¿ 0 (i = 1; : : : ; 5) and any matrices Xij and Yij

(i = 1; : : : ; 5; i¡ j6 5) such that the following LMIs are feasible:

=

11 12 13 14 ATS

T12 22 23 24 BTS

T13 T

23 33 34 0

T14 T

24 T34 44 CTS

SA SB 0 SC −S

¡ 0; (5)

" =

X11 X12 X13 X14 X15

X T12 X22 X23 X24 X25

X T13 X T

23 X33 X34 X35

X T14 X T

24 X T34 X44 X45

X T15 X T

25 X T35 X T

45 X55

¿ 0; (6)

60 Y. He et al. / Systems & Control Letters 51 (2004) 57–65

# =

Y11 Y12 X13 Y14 Y15

Y T12 Y22 Y23 Y24 Y25

Y T13 Y T

23 Y33 Y34 Y35

Y T14 Y T

24 Y T34 Y44 Y45

Y T15 Y T

25 Y T35 Y T

45 Y55

¿ 0; (7)

where

11 = PA + ATP + Q1 + Q2 + X15 + X T15 + Y15 + Y T

15 + �1X11 + �2Y11;

12 = PB− X15 + X T25 + Y T

25 + �1X12 + �2Y12;

13 = −ATPC + X T35 + Y T

35 − Y15 + �1X13 + �2Y13;

14 = X T45 + Y T

45 + �1X14 + �2Y14;

22 = −Q1 − X25 − X T25 + �1X22 + �2Y22;

23 = −BTPC − X T35 − Y25 + �1X23 + �2Y23;

24 = −X T45 + �1X24 + �2Y24;

33 = −Q2 − Y35 − Y T35 + �1X33 + �2Y33;

34 = −Y T45 + �1X34 + �2Y34;

44 = −R + �1X44 + �2Y44;

S = R + �1X55 + �2Y55: (8)

Proof. Choose the candidate Lyapunov functional to be

V (xt) := (Dxt)TPDxt +∫ t

t−�1

xT(s)Q1x(s) ds +∫ t

t−�2

xT(s)Q2x(s) ds

+∫ t

t−�2

xT(s)Rx(s) ds +∫ 0

−�1

∫ t

t+%xT(s)X55x(s) ds d% +

∫ 0

−�2

∫ t

t+%xT(s)Y55x(s) ds d%; (9)

where P=PT ¿ 0; Qi=QTi ¿ 0 (i=1; 2); R=RT ¿ 0 are weighting matrices, and X55=X T

55¿ 0; Y55=Y T55¿ 0.

All of these matrices need to be determined. Calculating the derivative of V (xt) along the solutions of �0

yields

V (xt) = 2(Dxt)TP[Ax(t) + Bx(t − �1)] + xT(t)Q1x(t) − xT(t − �1)Q1x(t − �1)

+ xT(t)Q2x(t) − xT(t − �2)Q2x(t − �2) + xT(t)Rx(t) − xT(t − �2)Rx(t − �2)

+ �1xT(t)X55x(t) −∫ t

t−�1

xT(s)X55x(s) ds + �2xT(t)Y55x(t) −∫ t

t−�2

xT(s)Y55x(s) ds: (10)

Y. He et al. / Systems & Control Letters 51 (2004) 57–65 61

Using the Leibniz–Newton formula, we can write

x(t − �1) = x(t) −∫ t

t−�1

x(s) ds; (11)

x(t − �2) = x(t) −∫ t

t−�2

x(s) ds: (12)

According to Eqs. (11) and (12), for any matrices Xi5; Yi5 (i = 1; : : : ; 4), the following equations hold:

2[xT(t)X15 +xT(t − �1)X25 +xT(t − �2)X35 + xT(t − �2)X45][x(t) − x(t − �1) −

∫ t

t−�1

x(s) ds]

= 0; (13)

2[xT(t)Y15 + xT(t − �1)Y25 + xT(t − �2)Y35 + xT(t − �2)Y45]

[x(t) − x(t − �2) −

∫ t

t−�2

x(s) ds

]= 0: (14)

On the other hand, for any appropriately dimensioned matrices Xij; Yij (i=1; : : : ; 4; i6 j6 4), the followingequation also holds:

x(t)

x(t − �1)

x(t − �2)

x(t − �2)

T

&11 &12 &13 &14

&T12 &22 &23 &24

&T13 &T

23 &33 &34

&T14 &T

24 &T34 &44

x(t)

x(t − �1)

x(t − �2)

x(t − �2)

= 0; (15)

where

&ij = �1(Xij − Xij) + �2(Yij − Yij); i = 1; : : : ; 4; i6 j6 4:

Then, we add the terms on the left sides of Eqs. (13)–(15) to V (xt); and consider the fact that, for anyr¿ 0 and any function f(t),∫ t

t−rf(t) ds = rf(t);

V (xt) can be expressed as follows:

V (xt) = zT1 (t)*z1(t) −

∫ t

t−�1

zT2 (t; s)"z2(t; s) ds−

∫ t

t−�2

zT2 (t; s)#z2(t; s) ds; (16)

where

z1(t) = [xT(t) xT(t − �1) xT(t − �2) xT(t − �2)]T; z2(t; s) = [zT1 (t) xT(s)]T;

* =

11 + ATSA 12 + ATSB 13 14 + ATSC

T12 + BTSA 22 + BTSB 23 24 + BTSC

T13 T

23 33 34

T14 + CTSA T

24 + CTSB T34 44 + CTSC

and "; #; ij (i = 1; : : : ; 4; i6 j6 4) and S are deBned in (5)–(8). If *¡ 0, "¿ 0 and #¿ 0, thenV (xt)¡ 0 for any z1(t) �= 0. Applying Schur complements, ¡ 0 implies that *¡ 0. So �0 is asymptoticallystable if the LMIs (5)–(7) are feasible.

62 Y. He et al. / Systems & Control Letters 51 (2004) 57–65

Remark 1. In Theorem 1, the relationships between the terms x(t− �1) and x(t)− ∫ tt−�1

x(s) ds, and x(t− �2)

and x(t) − ∫ tt−�2

x(s) ds have been considered through the free weighting matrices, Xi5 and Yi5 (i = 1; : : : ; 4);and the optimal weighting matrices can be selected by solving the LMIs (6) and (7). In contrast, previousmethods employed Bxed weighting matrices, which are not usually the optimal ones.

Remark 2. In previous methods, it is very hard to handle the case where the neutral and discrete delays arediCerent. As can be seen in Eq. (10), if we use the Leibniz–Newton formula to replace x(t−�1) and x(t−�2)in Dxt in the Brst term on the right side, a multiplication term involving two integrals appears, which is verydiJcult to deal with. To avoid this situation, x(t−�1) was replaced in some places using the Leibniz–Newtonformula, but was retained x(t − �2) in Dxt . Consequently, when that method is applied to a system withdiCerent neutral and discrete delays, the results are all discrete-delay-dependent and neutral-delay-independent.In Theorem 1, both x(t − �1) and x(t − �2) in Dxt are retained, but the relationships between the termsin the Leibniz–Newton formula are expressed in terms of free weighting matrices. This method overcomesthe diJculty that appears in previous studies when the neutral and discrete delays are diCerent. Thus, thecriterion in Theorem 1 includes information on the sizes of �1 and �2, which makes it both a neutral- and adiscrete-delay-dependent stability criterion. So, this method yields a less conservative criterion than previousdiscrete-delay-dependent and neutral-delay-independent criteria.

From Theorem 1, we obtain a delay-dependent criterion for the neutral system � with time-varying structureduncertainties.

Theorem 2. Given scalars �1 ¿ 0 and �2 ¿ 0, the neutral system � with time-varying structured uncertaintiesis robustly stable if the operator D is stable and there exist positive de>nite matrices P = PT ¿ 0, Qi =QT

i ¿ 0 (i=1; 2) and R=RT ¿ 0, non-negative de>nite matrices Xii =X Tii ¿ 0 and Yii =Y T

ii ¿ 0 (i=1; : : : ; 5),any matrices Xij and Yij (i = 1; : : : ; 5; i¡ j6 5) and a scalar +¿ 0 such that the LMIs (6), (7) and (17)are feasible.

11 + +ETa Ea 12 + +ET

a Eb 13 14 ATS PD

T12 + +ET

b Ea 22 + +ETb Eb 23 24 BTS 0

T13 T

23 33 34 0 −CTPD

T14 T

24 T34 44 CTS 0

SA SB 0 SC −S SD

DTP 0 −DTPC 0 DTS −+I

¡ 0; (17)

where ij (i = 1; : : : ; 4; i6 j6 4) and S are given in (8).

Proof. If A and B in (5) are replaced with A + DF(t)Ea and B + DF(t)Eb, respectively, then (5) for theuncertain system � is equivalent to the following condition:

+ -TdF(t)-e + -T

e FT(t)-d ¡ 0; (18)

where

-d = [ DTP 0 −DTPC 0 DTS ]; -e = [ Ea Eb 0 0 0 ]:

By Lemma 1, a necessary and suJcient condition for (18) for � is that there exists a +¿ 0 such that

+ +−1-Td-d + +-T

e -e ¡ 0: (19)

Applying Schur complements, we Bnd that (19) is equivalent to (17).

Y. He et al. / Systems & Control Letters 51 (2004) 57–65 63

Remark 3. The above results can easily be extended to a neutral system with multiple discrete- and neutral-delays.

4. Examples

This section presents some examples to illustrate the eCectiveness of the method described above.

Example 1 (Chen et al. [3], Fridman [5], Lien et al. [18]). Consider the stability of the nominal system �0

with

A =

[−0:9 0:2

0:1 −0:9

]; A1 =

[−1:1 −0:2

−0:1 −1:1

]; C =

[−0:2 0

0:2 −0:1

]:

The upper bounds on the delays that guarantee the stability of this system in [18,3,5] are �1 = �2 = 0:3,�1 =�2 =0:5658 and �1 =�2 =0:74, respectively. In contrast, solving LMIs (5)–(7) for �1 =�2, we obtained themaximum upper bounds on the allowable sizes to be �1 =�2 =1:6527, which are 450.9%, 192.1% and 123.3%larger than those in [18,3,5], respectively. This means that our method is less conservative than previousmethods. On the other hand, we also obtained the values for �1 �= �2. Table 1 lists the upper bounds on �1

that guarantee the stability of the system for values of �2 from 0:1 to 1:6. It can be seen that the upper boundon �1 decreases as �2 increases when �2 is small, but that �1 remains almost unchanged when �2¿ 1:2.

Example 2. Consider the robust stability of the system � with

A =

[−2 0

0 −0:9

]; B =

[−1 0

−1 −1

]; C =

[c 0

0 c

]; 06 c¡ 1;

D = I; Ea = Eb = 0:2I:

Table 2 shows some calculation results obtained from Theorem 2, and compares them to those ob-tained by the method of [9]. The values in the table are the maximum upper bounds on the delay �1.

Table 1Calculated allowable size of discrete delay �1 (Example 1)

�2 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9�1 1.7100 1.6987 1.6883 1.6792 1.6718 1.6664 1.6624 1.6591 1.6564

�2 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.6527 10000�1 1.6543 1.6531 1.6527 1.6527 1.6527 1.6527 1.6527 1.6527 1.6527

Table 2Calculated allowable size of discrete delay �1 (Example 2)

c

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4

Han’s paper [9] (�1 = �2) 1.77 1.63 1.48 1.33 1.16 0.98 0.79 0.59 0.37Theorem 2 (�1 = �2) 2.39 2.05 1.75 1.49 1.27 1.08 0.91 0.76 0.63Theorem 2 (�2 = 10000) 2.39 2.05 1.75 1.49 1.27 1.08 0.91 0.76 0.63Theorem 2 (�2 = 0:1) 2.39 2.25 2.11 1.96 1.81 1.66 1.50 1.33 1.16

64 Y. He et al. / Systems & Control Letters 51 (2004) 57–65

It is clear that our results are signiBcantly better when �1 = �2. On the other hand, when �1 �= �2, if themaximum upper bound on the delay �2 is small, then the maximum upper bound on �1 is larger than thatwhen �1 = �2. It can also be seen that the change in �2 has a big eCect on the upper bound on �1 when �2

is small, but only a small eCect when it is large.

5. Conclusion

This paper presents some new criteria for the delay-dependent robust stability of neutral systems withmixed delays and time-varying structured uncertainties. The criteria take into account the relationships betweenx(t−�1) and x(t)−∫ t

t−�1x(s) ds, x(t−�2) and x(t)−∫ t

t−�2x(s) ds. The free weighting matrices used to express

the relationships between, and the reciprocal in;uences of, these terms are selected by means of LMIs. Thecriteria arrived at in this paper are both neutral- and discrete-delay-dependent. Numerical examples illustrateboth the improvement that this method provides over previous methods, and also the reciprocal in;uencesbetween neutral delays and discrete delays.

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